3.3.1 Galerkin's Method for the Scalar Differential Operator

Equation (3.9) is written as

$$\int_{\mathcal{V}}N_i \left[\vec{\nabla}\cdot(\utilde{a}\cdot\ve... ...}\right] \mathrm{d}V - \int_{\mathcal{V}}N_if \mathrm{d}V = 0, i\in[1;n],$$ (3.14)

where Galerkin's method ($W_i = N_i$ ) and the scalar differential operator (3.2) are used. After applying the first scalar Green's theorem

$$\int_{\mathcal{V}}W\left[\vec{\nabla}\cdot(\utilde{a}\cdot\vec{u}... ...m{d}A - \int_{\mathcal{V}}\vec{\nabla}W\cdot\utilde{a}\cdot\vec{u} \mathrm{d}V$$ (3.15)

(3.14) is modified to read

$$\int_{\partial\mathcal{V}}N_i \vec{n}\cdot\utilde{a}\cdot\vec{\n... ...\tilde{u} \mathrm{d}V - \int_{\mathcal{V}}N_if \mathrm{d}V = 0, i\in[1;n].$$ (3.16)

Since $N_i$ vanishes on $\mathcal{A}_D$ for $i\in[1;n]$ the boundary integral from (3.16) reads

$$\int_{\partial\mathcal{V}}N_i \vec{n}\cdot\utilde{a}\cdot\vec{\n... ...\nabla}\tilde{u} \mathrm{d}A \simeq \int_{\mathcal{A}_N}N_i u_n \mathrm{d}A.$$ (3.17)

In (3.17) the conormal derivative $\vec{n}\cdot\utilde{a}\cdot\vec{\nabla}\tilde{u}$ corresponds with the Neumann boundary condition on $\mathcal{A}_N$ (3.4) [35,37]

$$\vec{n}\cdot\utilde{a}\cdot\vec{\nabla}\tilde{u} \simeq u_n = \vec{n}\cdot\utilde{a}\cdot\vec{\nabla}u \mathrm{on} \mathcal{A}_N.$$ (3.18)

Equation (3.16) leads with (3.5) and (3.17) to the linear equation system (3.12), where $[K]$ and $\{d\}$ are given by

$$\begin{split}K_{ij} & = - \int_{\mathcal{V}}\vec{\nabla}N_i\c... ..._N}N_i u_n \mathrm{d}A, i\in[1;n], j\in[1;n], \end{split}$$ (3.19)

with $\mathcal{L}[v]$ from (3.2).